
We note that (REF ) is closely related to the so-called U criterion [1]} widely used in computing the exceeding probability associated with a deterministic ItR. In general, the U criterion seeks the most `dangerous' point (i.e., point with maximum local variance), which in our problem corresponds to \(x^* = {\rm {argmax}}_{x} \; {\rm {std}}_{\omega _f,\omega _g}[\mathbb {P}(S(x, \omega | \omega _f, \omega _g) > \delta )]\)  (i.e., variance of (REF ) without considering the correlation at different \(x\)  and \(p_X(x)\) ). Therefore, our acquisition function in (REF ) can be considered as a weighted U criterion which incorporates the influence of the input distribution \(p_X(x)\)  in computing the variance of (REF ).
